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A note on Marcinkiewicz laws for strictly stationary ϕ -mixing sequences✩ Zbigniew S. Szewczak Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12/18, 87-100 Toruń, Poland

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Article history: Received 28 March 2011 Received in revised form 31 May 2011 Accepted 1 June 2011 Available online 12 June 2011

abstract A connection between boundedness in probability and laws of large numbers is established for ϕ -mixing strictly stationary sequences. © 2011 Elsevier B.V. All rights reserved.

MSC: 60F15 60F05 Keywords: Marcinkiewicz–Zygmund law Boundedness in probability ϕ -mixing Strictly stationary sequences Ibragimov’s conjecture

1. Introduction and results Let {Xk }k∈Z , Z = {. . . , −1, 0, 1, 2, . . .}, be a strictly stationary sequence defined on a probability space ∑(nΩ , F , P ) taking values on the real line R, and Fj k be the σ -field generated by Xj , Xj+1 , . . . , Xk , j, k ∈ Z, j ≤ k. Set Sn = k=1 Xk , n ∈ N = {1, 2, . . .} and define 0 ϕn = sup{|P (B|A) − P (B)|; P (A) > 0, A ∈ F−∞ , B ∈ Fn∞ }. If limn→∞ ϕn = 0, then {Xk } is said to be uniformly strong mixing or ϕ -mixing.

The main purpose of this note is to shed some light on the connection between boundedness in probability and laws of large numbers for dependent random variables. Recall that a sequence {Yn } is bounded in probability (b.i.p.) or stochastically bounded, if for each ϵ > 0 one can find M > 0 with supn P [|Yn | > M ] < ϵ . Our first result states the Marcinkiewicz–Zygmund law in terms of b.i.p. (cf. Marcinkiewicz and Zygmund, 1937, Théorème 9). 1

Theorem 1. Suppose r ∈ (1, 2) is a real number and {Xk } is a strictly stationary ϕ -mixing sequence. Then n− r Sn → 0 almost surely iff {n

− 1r

Sn } is b.i.p. and E [|X1 |r ] < ∞.

If moments do not exist, our next theorem gives a similar characterization (see also Szewczak, 2010, Theorems 1 and 2 and Remark 14). Define

cn = cn (r ) = sup x > 0; x−r E [|X1 |r I[|X1 |≤x] ] ≥

1

n

.

✩ Work supported in part by the Polish National Science Centre, Grant N N201 608740.

E-mail address: [email protected] 0167-7152/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2011.06.001

Z.S. Szewczak / Statistics and Probability Letters 81 (2011) 1738–1741

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Since x−r E [|X1 |r I[|X1 |≤x] ] → 0 as x → ∞, thus cn is finite and by Lemma 7 in Szewczak (2010) nE [|X1 |r I[|X1 |≤cn ] ] ∼ cnr . Theorem 2. Suppose {Xk } is a strictly stationary ϕ -mixing sequence and E [|X1 |r ] = ∞ for some real r ∈ (1, 2). If {cn−1 Sn } is b.i.p. and E [|X1 |r I[|X1 |≤x] ] is slowly varying in the sense of Karamata, then cn−1 Sn → 0 in probability. Conversely, if cn−1 Sn → 0 in probability and ϕ1 < 1, then E [|X1 |r I[|X1 |≤x] ] is slowly varying and E [X1 ] = 0. For {Xk } independent and identically distributed (i.i.d.) with E [|X1 |r I[|X1 |≤x] ] slowly varying, r ∈ (1, 2], {cn−1 Sn } is b.i.p. iff E [X1 ] = 0 (see Remark 1 and Lemma 2). In the case of sums of dependent random variables, a tool for verifying b.i.p. is von Bahr–Esseen’s inequality (cf. Bahr and Esseen, 1965, Theorem 2). Namely, if {Xk } is a (not necessarily stationary) random sequence that∑ E [|Xk |r ]⟨∞, k = 1, . . . , n, ⟩ and E [Xm+1 |Sm ] = 0 a.s., m ∈ N, 1 ≤ m ≤ n − 1, 1 ≤ r ≤ 2, then ∑n such n r E [| k=1 Xk | ] ≤ 2 k=1 E [|Xk |r ]. Note that the case r = 2, related to the Central Limit Theorem (CLT), is also included here. Another approach is the use of the truncation method together with the convergence rate of ϕn coefficients (cf. Szewczak, 2011, Corollary 1). Yet another inequality of interest is stated in Lemma 8.22 on p. 268 in Bradley, 2007 vol. I (see Remark 1). Unfortunately in our setting it is unknown if {cn−1 Sn } is b.i.p. or n−1 supn E [|Sn − E [Sn ]|r ] < ∞, r ∈ (1, 2], (cf. Bradley, 2007, vol. III, P2 Question, p. 457). This situation corresponds to the still unsolved Ibragimov’s conjecture (cf. Bradley, 2007, vol. III, P1 Ibragimov’s Conjecture, p. 457) and Iosifescu–Peligrad’s conjecture (cf. Bradley, 2007, vol. III, P3 Iosifescu’s Conjecture, p. 457; Peligrad, 1990, Conjecture 1.3) related to the CLT and its invariance principle, respectively. Therefore, it is natural to state the following conjecture, this time related to Marcinkiewicz laws of large numbers (see Lemma 2). Conjecture 1. Suppose {Xk } is a stationary random sequence which is ϕ -mixing and E [|X1 |r I[|X1 |≤x] ], r ∈ (1, 2], varies slowly. If E [X1 ] = 0, then {cn−1 Sn } is b.i.p. Remark 1. It is worth noting that under condition limn q∗n < 1 Conjecture 1 holds. Recall the definition of coefficient q∗n (cf. Bradley, 2007, vol. I, p. 249)

E

∑

Xj

j∈Q

∑

Xk

k∈S

, ∑ ∑ X X · j∈Q j k∈S k 2

∗

qn = sup

2

where this sup is taken over all pairs of nonempty, disjoint, finite sets Q , S ⊂ Z such that infj∈Q ,k∈S |j − k| ≥ n and ‖ · ‖2 is L2 norm. Since E [|X1 |r I[|X1 |≤x] ] varies slowly, lim ncn−1 E [|X1 |I[|X1 |>cn ] ] = lim n

n

xr −1 E [|X1 |I[|X |>x] ] 1 E [|X1 |r I[|X |≤x] ]

ncn−1 E [|X1 |I[|X1 |>cn ] ] n E cnr

→ 0 as x → ∞ (cf. Feller, 1971, Theorem 2, p. 283), therefore

1

[|X1 |r I[|X1 |≤cn ] ]

= 0.

On the other hand, by Lemma 8.22 on p. 268 in Bradley (2007), vol. I

2 n − 1 + q∗m n 1 + q∗m n cn−2 E E [X12 I[|X1 |≤cn ] ] ≤ 4m E [|X1 |r I[|X1 |≤cn ] ], Xk I[|Xk |≤cn ] − E [Xk I[|Xk |≤cn ] ] ≤ 4m ∗ 2 ∗ cr 1 − q c 1 − q m n m n k=1 for m such that q∗m < 1 and n > m. Since cnr E [|X1 |r I[|X1 |≤cn ] ] is bounded, therefore {cn−1 Sn } is b.i.p. by the Chebyshev inequality n and the truncation argument. Thus, if Bradley’s conjecture (cf. Bradley, 2007, vol. III, P10 Conjecture, p. 461) turns out to be true, then Conjecture 1 would follow. The note is organized in such a way that in Section 2 there are some auxiliary inequalities required for the proofs in Section 3. 2. Auxiliary inequalities Let {Xk } be a (not necessarily strictly stationary) random sequence and n > m, i ∈ N, p, s, t , u > 0. Recall the following extension of the Hoffmann-Jørgensen inequality from Szewczak (2010). P [ max |Sk | > s + 2t + u; m · max |Xi | ≤ u] ≤ (ϕm + P [ max |Sk | > s]) · P [ max 1≤k≤n

1≤i≤n

m≤k≤n

1≤k≤n−m

|Sk | > t ],

(2.1)

k where ϕm = supk∈Z {|P (B|A) − P (B)|; A ∈ F−∞ , B ∈ Fk∞ +m }. Similarly, in Szewczak (2010) we gave a dependent version of Lévy’s inequality (cf. Marcinkiewicz, 1939, Théorème 3), if the laws of reversed sums L(Sn − Sk ) are symmetric for n > k ≥ 1

1 2

− ϕm · P [

max 1≤k≤n−m+1

|Sn | > t ] ≤ P [|Sn | + (m − 1) · max |Xi | > t ]. 1≤i≤n

(2.2)

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The following moment inequality can be derived from (2.1) (cf. Szewczak, 2010) E [ max |Sk |p ] ≤ 1≤k≤n

4p 1−τ

· (mp E [ max |Xi |p ] + tτp ),

(2.3)

1≤i≤n

where

tτ = inf t > 0; ϕm + P [ max |Sk | > t ] ≤ m≤k≤n

τ

4p

,

τ ∈ (0, 1).

Since inequalities (2.1)–(2.3) include the term Mn = max1≤i≤n |Xi | the following inequality for strictly stationary sequence {Xk } is useful because it allows replacing dependent maxima satisfying Doeblin’s condition (i.e. ϕm < 1) by the maxima of its associated sequence {Xk∗ } (i.e. the i.i.d. sequence with L(X1 ) = L(X1∗ )):

(1 − ϕm ) · P [M⌊∗ n ⌋ > t ] ≤ P [Mn > t ] ≤ m · (1 + ϕm ) · P [M⌊∗ n ⌋+1 > t ], m

m

t ≥ 0, n ≥ 1,

(2.4)

where Mn∗ = max1≤i≤n |Xi∗ | (cf. Peligrad, 1990, p. 298). Recall also von Bahr–Esseen’s symmetrization inequality (cf. Bahr and Esseen, 1965, Lemma 4) 1 2

E [| X |r ] ≤ E [|X − E [X ]|r ] ≤ E [| X |r ],

r ∈ [1, 2],

(2.5)

where X = X − X˜ and X˜ is an independent copy of X . 3. Proofs Theorems 1 and 2 are implied respectively by Theorem 1 in Szewczak (2011), as well as Theorems 1 and 2 in Szewczak (2010) and the following two lemmas (cf. Szewczak, 2010, Remark 14). Lemma 1. Suppose {Xk } is a strictly stationary ϕ -mixing sequence, r ∈ (1, 2] and E [|X1 |r I[|X1 |≤x] ] varies slowly. If {cn−1 Sn } is b.i.p., then

r n − (Xk I[|Xk |≤cn ] − E [X1 I[|X1 |≤cn ] ]) < ∞. sup cn E k=1 n −r

(3.6)

r Proof of Lemma 1. Set Sn′ = k=1 Xk I[|Xk |≤cn ] . Since E [|X1 | I[|X1 |≤x] ] varies slowly, thus we have nP [|X1 | > ϵ cn ] → 0, ϵ > 0 (cf. Feller, 1971, Theorem 2, p. 283). Hence

∑n

cn−1 max |Xi | → 0, 1≤i≤n

in probability.

Therefore, {cn−1 Sn′ } is b.i.p. Set Sn′ =

∑n

k=1

(3.7)

Xk I[|Xk |≤cn ] − X˜ k I[|X˜ k |≤cn ] , where {X˜ k } is an independent copy of {Xk }. Hence {cn−1 Sn′ }

is b.i.p. too. Choose m such that ϕm < 1 − √1 so that ϕm = ϕm ({ Xk }) < 2

1 2

(cf. Bradley, 2007, vol. I, Theorem 6.6, p.199). Now

(2.2) and (3.7) yield that {max1≤k≤n cn |Sk′ |} is b.i.p. Moreover, by (2.4) n limn cn−r E [ max |Xi |r I[|Xi |≤cn ] ] ≤ 2mlimn cn−r E [|X1 |r I[|X1 |≤cn ] ] ≤ 2 −1

1≤i≤n

m

thus sup cn−r E [ max |Xi I[|Xi |≤cn ] − X˜ i I[|X˜ i |≤cn ] |r ] < ∞. n

1≤i≤n

Therefore, by (2.3) we get E [max1≤k≤n cn−r |Sk′ |r ] < C for some C < ∞ and every n ∈ N. Consequently, condition (3.6) follows from (2.5). This completes the proof. It is interesting that the converse of Conjecture 1 holds. Lemma 2. Suppose {Xk } is a strictly stationary ϕ -mixing sequence, r ∈ (1, 2] and E [|X1 |r I[|X1 |≤x] ] varies slowly. If {cn−1 Sn } is b.i.p., then E [X1 ] = 0. Proof of Lemma 2. Since r > 1, and E [|X1 |r I[|X1 |≤x] ] varies slowly, thus E |X1 | < ∞. By the proof of Lemma 1 we see that {cn−1 Sn′ } is b.i.p. Moreover, by the Markov inequality and (3.6), the sequence {cn−1 (Sn′ − nE [X1 I[|X1 |≤cn ] ])} is also b.i.p. so that

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{cn−1 nE [X1 I[|X1 |≤cn ] ]} is b.i.p. too. Furthermore, by a variant of Karamata’s theorem (cf. Feller, 1971, Theorem 2, p. 283) cnr −1

E [|X1 |I[|X1 |>cn ] ] E [|X1 |r I[|X1 |≤cn ] ]

→ 0 as n → ∞.

(3.8)

Since nE [|X1 |r I[|X1 |≤cn ] ] ∼ cnr , thus (3.8) yields cn−1 nE [|X1 |I[|X1 |>cn ] ] → 0

as n → ∞.

−1

Therefore, {cn n|E [X1 ]|} is a bounded sequence. On the other hand, the slow variation of E [|X1 |r I[|X1 |≤x] ] entails cnr n−r ∼ n1−r E [|X1 |r I[|X1 |≤cn ] ] → 0

as n → ∞.

Thus, the only possibility is E [X1 ] = 0 and Lemma 2 follows.

Acknowledgments The author thanks the anonymous referee and Andrzej Szewczak for constructive remarks and suggestions that improved the presentation of the paper. References Bahr, B.von, Esseen, C.-G., 1965. Inequalities for the r–th absolute moment of a sum of random variables 1 ≤ r ≤ 2. Ann. Math. Stat. 36, 299–303. Bradley, R.C., 2007. Introduction to Strong Mixing Conditions, vol. I–III. Kendrick Press. Feller, W., 1971. An Introduction to Probability Theory and Its Applications, 2nd ed.. vol. II. Wiley, New York. Marcinkiewicz, J., 1939. Quelques théorèmes de la théorie des probabilités. Prace Towarzystwa Przyjaciół Nauk w Wilnie. Wydział Nauk Matematycznych i Przyrodniczych. Travaux de la Société des Sciences et de Lettres de Wilno. Classe des sciences mathématiques et naturelles T. XIII. Bull. Sém. Math. Univ. Wilno 2, 22–34 (in: J. Marcinkiewicz, Collected Papers, PWN, Warszawa, 1964, pp. 566–579). Marcinkiewicz, J., Zygmund, A., 1937. Sur les fonctions indépendantes. Fund. Math. 29, 60–90. Peligrad, M., 1990. On Ibragimov–Iosifescu conjecture for ϕ –mixing sequences. Stochastic Process. Appl. 35, 293–308. Szewczak, Z.S., 2010. Marcinkiewicz laws with infinite moments. Acta Math. Hungar. 127, 64–84. Szewczak, Z.S., 2011. On Marcinkiewicz–Zygmund laws. J. Math. Anal. Appl. 375, 738–744.